Optimal. Leaf size=174 \[ \frac {2 \sqrt {2 \pi } b^{3/2} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 \sqrt {2 \pi } b^{3/2} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {8 b \sinh (a+b x) \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3314, 32, 3312, 3307, 2180, 2204, 2205} \[ \frac {2 \sqrt {2 \pi } b^{3/2} e^{\frac {2 b c}{d}-2 a} \text {Erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 \sqrt {2 \pi } b^{3/2} e^{2 a-\frac {2 b c}{d}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {8 b \sinh (a+b x) \cosh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 3314
Rubi steps
\begin {align*} \int \frac {\sinh ^2(a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {1}{\sqrt {c+d x}} \, dx}{3 d^2}+\frac {\left (16 b^2\right ) \int \frac {\sinh ^2(a+b x)}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=\frac {16 b^2 \sqrt {c+d x}}{3 d^3}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}-\frac {\left (16 b^2\right ) \int \left (\frac {1}{2 \sqrt {c+d x}}-\frac {\cosh (2 a+2 b x)}{2 \sqrt {c+d x}}\right ) \, dx}{3 d^2}\\ &=-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {\cosh (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (4 b^2\right ) \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{3 d^2}+\frac {\left (4 b^2\right ) \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (8 b^2\right ) \operatorname {Subst}\left (\int e^{i \left (2 i a-\frac {2 i b c}{d}\right )-\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{3 d^3}+\frac {\left (8 b^2\right ) \operatorname {Subst}\left (\int e^{-i \left (2 i a-\frac {2 i b c}{d}\right )+\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{3 d^3}\\ &=\frac {2 b^{3/2} e^{-2 a+\frac {2 b c}{d}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}+\frac {2 b^{3/2} e^{2 a-\frac {2 b c}{d}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{3 d^{5/2}}-\frac {8 b \cosh (a+b x) \sinh (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sinh ^2(a+b x)}{3 d (c+d x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.12, size = 156, normalized size = 0.90 \[ -\frac {2 e^{-2 \left (a+\frac {b c}{d}\right )} \left (e^{2 \left (a+\frac {b c}{d}\right )} \left (2 b (c+d x) \sinh (2 (a+b x))+d \sinh ^2(a+b x)\right )+\sqrt {2} e^{4 a} d \left (-\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 b (c+d x)}{d}\right )+\sqrt {2} d e^{\frac {4 b c}{d}} \left (\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {2 b (c+d x)}{d}\right )\right )}{3 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.78, size = 864, normalized size = 4.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}\left (b x +a \right )}{\left (d x +c \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 118, normalized size = 0.68 \[ -\frac {\frac {3 \, \sqrt {2} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {3}{2}, \frac {2 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}} + \frac {3 \, \sqrt {2} \left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {3}{2}, -\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}} - \frac {2}{{\left (d x + c\right )}^{\frac {3}{2}}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________